Number Operations & Proportional Reasoning K-8 Teachers

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por Melody Wolen

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por David Lanier

Computing for Humans

por Dénes Harmath

Sorting Algorithms

por Katie Delisle

Week 16

12/02 : FINAL

11/30: Homework review & Test Review

Week 15

11/25: Thanksgiving Break

Day 1: 11/23 - Integers - Integers: Positive & negative numbers "chip method" 🔴 = negative 🟡 = positive 🟡 🔴 together = zero pair 5 + 1 = +6 🟡 🟡 🟡 🟡 🟡 + 🟡 = 6 5 + -1= +4 🟡🟡🟡🟡 🟡 + 🔴 = 4 -5 + -1 = -6 🔴🔴🔴🔴🔴 + 🔴 = -6 -5 + 1 = -4 🔴🔴🔴🔴 🔴 + 🟡 = -4

Week 14

Day2: 11/18 - Percentages - is : = of: x (multiplication sign) what : n, x (variables) change % to decimal EX: 72% → 0.72 10 % off this means that for every $100, you get 10% off 1/10 = 0.1 = 10% 21/100 = 0.21 =21% * Equivalent Fractions for Conversions * 2/5 = 40/100 = 0.41 = 41% * Divide to Convert * 1/8 = 1 ÷ 8 = 0.125 = 13% 0.125 = terminating decimal 5/6 = 5 ÷ 6 = 0.833333 (repeating decimal) = 83% π (pi) = 3.14... (irrational number) pi is just like love, irrational and never ending Practice: 8 is what % of 22 8 = % x 22 22a = 8 22/22 a = 8/22 a= .36666 = 37%

Day 1: 11/16 - Decimals - 3 7 5 . ↓ ↓ ↓ ↓ ↓ ones ↓ tens hundreds one-to- ten relationship ← # get bigger (10 times) → # gets smaller (10 times) whole/ 1 unit = 100 single cubes (imagine a whole as a ten by ten row unit) 10x 10s = 1 unit 1 x10 = 1/10 (part of a unit) 1 x 1 = 1/100 Day 1: 11/16 - Decimals - 3 7 5 . 2 5 3 (# after decimal=parts of a unit) ↓ ↓ ↓ ↓ ↓ thousandths ↓ hundredths tenths 3 7 5 . 2 5 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 1¢ ↓ ↓ ↓ 10¢ ↓ ↓ $1 ↓ $10 $100 ← x 10 ÷ 10 → What does a decimal do? It separates the parts from the whole # in between decimals 0.3 & 0.4 → 0.35 0.32 & 0.33 → 0.321, 0.322...

Week 13

11/11: VETERANS DAY - NO SCHOOL

11/09: (absent) TEST OVERVIEW

Week 12

11/04: CLASS CANCELLED

11/02: TEST #2

Week 11

Day 2: 10/28 - HW & TEST Review/ Mind-map in Class 1. 1/3 cups of sugar needed to make 2 loaves How many cups of sugar are needed for 3 loaves? - 1/2 a cup of oil to make 3 loaves 2. Sometimes you have to simplify when adding fractions with a least common denominator ** Review for test

Day 1 : 10/26 - Fraction Problems in class - 1. 3/4 cups per serving 2 and 2/3 needs to prepare, how many cups of oil will she need? 2 cups 2. 3/4 pizza 1/2 ate How much did he eat? He ate 3/8 of the pizza 3. if 4 * = 2/7 of the whole ** ** ** ** ** ** ** 2**= 1/7 whole of 14 stars total

Week 10

Day 2: 10/21 - In Class Fraction Problems - 1. Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get? Jim= 12 Ken and Len = 16 each Max = 4 total: 48 2. Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of the bars. Then Ken took 1/3 of the remaining bars. Next, Len took 1/3 of the remaining bars, and Max took the remaining 8 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get? How is this problem (with regards to fractions) different from Problem 1? Jim= 6 Ken= 6 Len= 4 Max= 8 3. 3/4 of a pie in refrigerator John ate 2/3 left over How much did he eat? John ate 1/2 of the pie 4. Three-fourths of the class are girls. Two-thirds of the girls have black hair. What fraction of the class is female and dark-haired? half of the class have dark hair **Look back at notes

Day 1: 10/ 19 - Fractions Continued- identity property of multiplication: 2/3 + 2/2 = 4/6 25/100 ÷ 25/25 = 1/4 ** find common factor for both 25 and 100 a/a = 1 x/x= a 28ab2/42a2b2 = 7*4*a*b*b / 7*6*a*a*b*b = 4/6a = 2/3a Adding Fractions with common denominator 3/12 + 2/12 =5/12 1/4+3/4= 4/4 =1 Adding Fractions with different denominator 1/4 + 1/6 l l v v 3/12 + 2/12 =6/12 = 1/2 - Find the least common multiple by finding the multiple ** Yard example (look back at notes)

Week 9

10/ 14 - FRACTIONS - Fraction: is a part of a whole; it is a symbol that represents a part to a whole - quotient -part of a whole - division - ratio (to be a fraction - part to whole not part to part) EX: 3/10 3= part 10 = whole 30 balloons 3 red 27 blue 3/30 27/30 How you can demonstrate fractions to students: - surface area (region) -shading - length - folding a paper - number line - line segment Mistakes made by students: 1/2 < 1/8 2/3 + 1/5 = 3/8 Practice: 4/7 or 5/7 4/5 or 4/9 3/7 or 5/ 8 9/10 or 3/4 Wholes: 8/8, 6/6, 4/4, 3/3, 1/2 what did you discover? When you use all the pieces you have 1 whole numerator = denominator = 1 whole

10/11 NO SCHOOL

Week 8

Day 2: 10/07 - Prime Numbers - Prime #s: 2,3 5, 7, 11,13,17,19,23, 29,31 37, 41,43,47,53,59 GCF: greatest common factor LCM : least common multiple List Method: 24: 1,2,3,4,6,8,12,24 36:1,2,3,4,6,9,18,36 GCF: 12 **you use common factors when you simplify EX: 25/100 ÷ 5/5 = 5/20 ÷ 5/5 = 1/4 vs. if you use your GCF 25/100 ÷ 25/25 = 1/4 24: 24, 48, 72, 96 36: 36,72 LCM: 72 **improper fraction 10/15 + 9/15 = 19/15 Prime Factorization : 20:1,2,4,5,10 20 20 / \ / \ 4 5 2 10 /\ /\ 2 2 2 5 2 x 2 x 5 = 22 x 5 **If you take your found GCF and multiply by the remaining #s = LCM EX: 24: 2 x 2 x 2 x 3 36: 2 x 2 x 3 x 3 GCF: 12 after doing prime tree: 24: (23 x 3) 36: (22 x 32) 12 x 2 x 3 = 72 ** list method is preferred for students due to the fact that if they mess up trying to find the GCF/LCM they will mess up their GCF/ LCM (both numbers)**

Day 1: 10/05 - Divisibility and Factors - Divisibility: “a” is divisible by “b”, if there is a number “c” that meets the requirements EX: 10 is divisible by 5 b x a = c 5 x 2 = 10 Important Terminology: *10 is divisible by 5 or 5 divides 10 *5 is a divisor of 10 *5 is a factor of 10 *10 is. multiple of 5 Divisibility Rules: —————————- Ending By 2: 0,2,4,6,8 By 5: 0,5 By 10: 0 (a number is divisible by 2 if it ends in a 0,2,4,6,8; 5 is divisible if a number ends in 0,5; 10 is divisible by a number if it ends in 0) Sum of Digits: By 3: *when the sum of the digits are divisible by 3* EX: 543 5+4+3 = 12/3 = 4 By 9: * when the sum of the digits are divisible by 9 ; just like #3* Other: by 6: if it is divisible by both 2&3 EX: 3702 Last Digits: by 4: if the last two digits are divisible by 4 EX: 3,728 28/4 by 8: if the last three digits are divisible by 8 Special Numbers: by 7: EX: 826 step 1 - (double last digit) 6x2=12 step 2 - (take your first two digits from your given number and and subtract it from step 1) 82-12 = 70 *70 is divisible by * *if there is a bigger number repeat the first two steps* Chop-Off: by 11: -Chop off last two digit numbers - add them to the remaining number(s) - repeat EX: 29,194 291 +94 = 385 ; 85+3 = 88/11=8 Factors: 28: 1,2,4,7,14,28 42:1,2,3,6,7,14,21,42 39: 1,3,13,39 91: 1,7,21,91 84: 1,2,3,4,6,7,12,14,21,28,42,84 60: 1,2,3,4,5,6,10,12,15,20,30,60 \ / composite numbers Prime numbers: *have two factors* when it has factors of 1 and itself Prime #s 1-60: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 0: not a prime - not composite 1: not prime- not composite 0: additive identity element 1: multiplicative identity element

Week 6

Day 2: 9/23 - Algorithms vs. Strategies Algorithm - when you have pencils and paper Strategies - things we do in our brain when we don't have a pencil and paper by us - Mental Strategies - Addition: 1. Left to Right (Front-end Addition) 347 + 129 Head: 300 + 100 = 400 40+20 = 60 7+9 = 16 -------------------------- = 476 2. Compensation 67+29 Head: 67+30 = 97 -1 --------------------- 96 3. Compatible numbers 130 + 50 +70 +20+ 50 Head: 50+50 + 70 + 130 + 20 l l l l l l 100 200 = 320 4. Breaking Up & Bridging 67 + 36 Head: 67 + 36 l l _l_ l l 30 6 67 + 30 = 97 +6 ---------------- 103 Subtraction: 1. L -> R a) 47-32 = 15 Head: 40-30 = 10 7-2 = 5 ----------------- = 15 2. Breaking up and Bridging 67 - 36 = 31 Head: 67 - 36 l l l l l 30 6 67 - 30 = 37 -6 ---------------- 32 3. Compensation 47 - 29 = 18 Head: 47-30 = 17 +1 ——————————----- 18 4. Compatible Numbers: 170 - 50 - 30 - 50 = Head: 170+ 30 = 140 50+50 = 100 140-100= 40 Multiplication: 1. Compatible numbers 2 x 9 x 5 x 20 x 5 Head: 5 x 2 = 10 20 x 5 = 100 9 x 1000 = 9,000 2. L -> R 3x 123 Head: 3x(100+20+3) 300+60+9 = 369 Division: 1. Compatible Numbers 105 ÷ 3 Head: 150 ÷ 3 = l__l _l__ l l 90 +15 90/3 = 30 > 35 15/3 = 5 TEST REVIEW***

Day 1: 9/21 Class Cencelled

Week 7

09/30 *TEST OVERVIEW*

09/28 *TEST 1*

Week 5

Day 2 (9/16) - Division 4 r 4 5/ 24 - 20 --------- 4 4 = quotient 4 = aka 4/5 5 = divisor 24 = dividend *division symbol* = vinculum 11 ÷ 3 = 3 r 2 11-3 = 8 8-3 = 5 5-3 = r 2 Standard Algorithm 158 r 1 3 / 475 - 3↓↓ --------- 17 ↓ - 15 ↓ ---------- 25 - 24 ---------- 01 Standard Alg. Emphasis on Place Value 158 r 1 3 / 475 - 300 --------- 175 - 150 ---------- 25 - 24 ---------- 01 **separation of the numbers into three boxes*** Alternate Algorithm 197 cookies “You work at a factory and your boss tells you to put 13 cookies in bags” 12 r 5 16 / 197 - 160 10 bags ---------- + 2 bags 37 ------------ - 32 12 bags ---------- 5 https://www.youtube.com/watch?v=KGMf314LUc0

Day 1 (9/14) - Multiplication Grouping 3 groups of 2 → 3x 2 → 2+2+2 → 2, 4, 6 4 groups of 3 → 4x3 → 3+3+3+3 →3,6,9,12 **number line example ** **Clock example** **array 3 x 8 *** Commutative Property of Multiplication 3 x 8 = 8 x 3 Identity Property of Multiplication a x 1 = 1 Zero Property Of Multiplication a x 0 = 0 Associative Property of Multiplication (a x b ) x c = a x (b x c) Distributive Property of Multiplication 3 x (5 + 2) = (3 x 5 ) + (3 x 2) ↓ ↓ partial product → When you multiply a number by a sum, the product remains the same Multiplication Algorithms 1. Standard Algorithm of Multiplication → regular way of multiplying in the U.S 2. Place Value 23 4x 3 = 12 x 14 4 x 20 = 80 -------- 10 x 3 = 30 322 10 x 20 = 200 3. Expanded Notation of Multiplication 23 = 20 + 3 x 14 = 10 + 4 -------- ---------- 90 + 2 200 + 30 + 0 (+100) ---------------------- 300 + 20 + 2 4. Lattice Algorithm of Multiplication → Lattice box → apply the numbers around the box → solve the problem within the box https://www.youtube.com/watch?v=FJ5qLWP3Fqo

Week 4

Day 2 (9/9) - Subtraction Subtraction - known as "take away", "missing addend" , - "I have four cookies, how many more do I need to get from the cookie jar so I have a total of 7?" 4 + 🍪 = 7 -4 -4 --------------- 🍪 = 3 - "I have 4 cookies, Skyler has 3 cookies, how many more cookies do I have?" 34 → minuend - 12 → subtrahend ------- 22 → difference 1. American Standard Algorithm - Subtract as you were taught (USA) 2. European- Mexican Algorithm - subtract (R to L) - borrow from the bottom row numbers but make those a digit bigger 3. Reverse Indian - Subtract (L to R) - borrow from the numbers you have acquired from your answer 4. Left to Right - focuses on place value - like the Reverse Indian but you demonstrate the actual place values vs just numbers 5. Expanded Notation https://www.youtube.com/watch?v=ShCq1BVVbQ0

Day 1 (9/7) - Addition Properties of addition: 1. Identity Property of Addition a + 0 = a 2. Commutative Property of Addition a + b = b + a 3. Associative Property of Addition (a+b) +c = a + (b+c) Addition : 1. Standard American Algorithm 576 + 279 ---------- - Start from the one's place (R to L) - 6+ 9 =15 - write the 5 in the ones;add the 1 up of the 7 2. Partial Sums Algorithm - (R to L) -add starting at the ones - when the total of the row equals more than the place value, take it down to the total vs putting it above the next place value 3. Partial Sums - Emphasis on Place Value -(R to L) - add starting at the ones - when adding, place the total below the equal sign but put the actual total instead of putting it above the next place value 4. Left to Right Algorithm -(L to R) - add in an expanded form kind of way 5. Expanded Notation - (R to L) - add in expanded notation terms - Place value explicit 6. Lattice Algorithm - (R to L) - draw a lattice box and add https://www.youtube.com/watch?v=kTBZdIXizDQ

Week 3

Day 2 (9/2) - Base 2 Base 2 - Binary System (Computers Use) Digits Used: (0,1) 1101₂ = 27 = (1 x 2^4) + (1x 2^3) + (0 x 2^2) + (1x 2^1) + (1x 2^0) =(1x 16) + (1x8) + (0x4) + (1x2) + (1x1) = 16+ 8 + 0+ 2+1 = 27 1111₂ = 15 =(1x 2^3) + (1x 2^2) +(1 x2^1) + (1x 2^0) =(1x 8) + (1x4 )+ (1x2) + (1) =8+4+2+1 =15 ** Base 5 Practice**

Day 1 (8/31)- intro to Base 10, 5, 3 -Base 10- Digits used: 0,1,2,3,4,5,6,7,8,9 numeral system: base- 10 system expanded notation = powers of ten 5, 375.32 5 - thousands , 3 - hundreds 7- tens 5 - ones . 3 - tenths 2- hundredths $275.25 2 - hundred dollar bills 7- ten dollar bills 5 - ones dollar bills . 2 - dimes 5- pennies Expanded Notation: 273 200+70+3 = (2x100) + (7x10) + (3x1) -Base 5 - Digits Used: 0, 1, 2, 3, 4 43₅ = 23 (4 x 5) + (3x1) (4 x 5^1) + (3 x 5^0) 20 + 3 = 23 11.1₅ = =(1 x 5) + (1x1) + (1x 1/5) =(1x 5^1) + (1x 5^0) + (1x 1/5) 5 +1+ 1/5 = 6 1/5 - Base 3 - Digits Used: 0, 1, 2 111₃ = 13 (1 x 9) + (1 x 3) + (1 x 1) =(1 x 3^2) + (1 x 3^1) + (1 x 3^0) 9 + 3+ 1 = 13 211₃ = (2 x 3^2) + (1x 3^1) + (1 x 3^0) =(2x9) + (1x3) + (1) = 18 +3 +1 = 22 https://www.youtube.com/watch?v=jdfrwxUkXOk

Week 2

Day 2 (8/26) - Problem Solving A student was making sandwiches 13 of the sandwiches had a slice of cheese. 14 had a slice of salami. 13 had a slice of tomato. 8 had a slice of cheese & a slice of tomato. Only 3 had a just a slice of salami. 5 had a slice of tomato, a slice of cheese, & a slice of salami. 8 had a slice of tomato & a slice of salami. How many sandwiches did the student make? In our groups we attempted to figure out the best way to solve this problem. Then we shared how we came up with the answer. After sharing, the instructor demonstrated how to solve the problem with a ven diagram. The total was : 21 sandwiches

Day 1(8/24) - Problem Solving We were introduced to the use of sticks and cubes to problem solve. Students learn best in different and unique ways. When problem solving, we have to 1. understand the problem 2. devise a plan (strategize) 3. carry out the plan 4. reflect ( does it work?) Problem Solving Strategies: 1. Look for a pattern 2. Examine a related Problem 3. Make a diagram 4. Work backwards 5. Use guess and Check 6. Write and equation https://www.youtube.com/watch?v=J3GGx9wy07w

Week 1

Day 1 (8/19) - We went over the syllabus and Mind-map assignment https://player.mediaamp.io/p/U8-EDC/qQivF4esrENw/embed/select/media/cO8kAp7DPj_c?form=html

Number Operations & Proportional Reasoning K-8 Teachers